A Jordan curve is a plane curve which is topologically equivalent to (a homeomorphic image of) the unit circle, i.e., it is simple and closed.

It is not known if every Jordan curve contains all four polygon vertices of some square, but it has been proven true for "sufficiently smooth" curves and closed convex curves (Schnirelman 1944; Steinhaus 1999, p. 104). For every triangle and Jordan curve , has an inscribed triangle similar to .

Krantz, S. G. "Closed Curves." §2.1.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 19-20, 1999.

Schnirelman, L. G. "On Certain Geometrical Properties of Closed Curves." Uspehi Matem. Nauk 10, 34-44, 1944.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

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Weisstein, Eric W. "Jordan Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JordanCurve.html